The Three-Jet Rate in $e^{+}{e}^{-}$ Annihilation

Progress has been made on the calculation of $R_3$, the three-jet rate in $e^{+}{e}^{-}$ annihilation, in the $k_{\mathrm{\perp }}$ (Durham) scheme. Using the coherent branching formalism \cite{b,c,d}, an explicit expression for $R_3$ is calculated. In this, leading and next-to-leading large logarithms (LL and NLL) are resummed to all orders in QCD perturbation theory. In addition to exponentials an error function is involved.